WASHINGTON, May 7 (Reuters) - Kentucky Senator Rand Paul bet on a long-shot and lost in North Carolina's contentious Republican Senate primary, raising new questions about the Tea Party hero's bid to woo the party establishment on his way to a 2016 White House race.

While many Republicans celebrated Tuesday's Senate primary win by North Carolina House Speaker Thom Tillis, Paul scrambled to save face after the candidate he backed, Greg Brannon, finished a distant second in the battle for the right to challenge vulnerable Democratic Senator Kay Hagan.

"Now that the primary is over, it is time for our side to unite to defeat the Democrat who cast the deciding vote for Obamacare, Kay Hagan," Paul said in a statement.

Paul had put his growing party clout on the line at an appearance on Monday with Brannon, whom he praised as a "hero" despite his history of provocative statements like calling President Barack Obama a fascist and contending the constitutional right to bear arms extends to nuclear weapons.

Given that Brannon was precisely the type of risky and divisive candidate the party's mainstream leaders are striving to avoid this year, Paul's endorsement was a mystery to many Republicans and a sign of the difficult line he walks in courting the establishment without alienating his libertarian base.

"He put himself in the wrong camp in North Carolina, and that was a big mistake. When you are running for president, you don't ever want to be with the losers, even if you are making an ideological pick," Republican strategist Ron Bonjean said.

While keeping one foot in the Tea Party camp, Paul also has backed fellow Kentucky Senator Mitch McConnell, a mainstay of the party establishment as the leader of Senate Republicans, in his May 20 primary against a Tea Party opponent.

That follows months of wooing the party's establishment donors while touting his ability to appeal to young people and minorities. At last weekend's Kentucky Derby, Paul courted media magnate Rupert Murdoch, whose News Corp. controls the two most powerful mainstream media voices in Republican politics - Fox News and the Wall Street Journal's editorial pages.

He also has been on a busy national speaking tour that took him before audiences Republicans typically avoid in places like the liberal bastion of Berkeley, California, and historically black Howard University in Washington.

A RISING PROFILE

That groundwork for 2016 has paid off with a rising profile and a jump into the top tier of potential Republican White House candidates in recent opinion polls.

At the same time, Paul has drawn flak from some social conservatives unhappy with his hands-off views on abortion and gay marriage and from foreign policy hawks concerned about his opposition to a more active U.S. global presence.

Steve Deace, a conservative talk radio host in Iowa, said Paul has been "trying to thread the needle to appeal to all sides."

Deace said social conservatives, a powerful bloc in Iowa's Republican nominating contest that kicks off the race, would not back a candidate who said, as Paul did recently, that he would not push to change abortion laws until public opinion shifts.

"Whatever support he has in the Iowa caucuses right now is the most he is ever going to have," Deace said. "I think he wants to be president too bad."

But Trey Greyson, who lost to Paul in Kentucky's 2010 Senate Republican primary and is now director of Harvard University's Institute of Politics, said Paul's opposition to foreign intervention and to the National Security Agency's vast surveillance was becoming a more mainstream Republican stance.

"His views have become more popular inside the party in recent years. On the NSA stuff, Rand is right where the party is," Greyson said, adding some of the recent criticism is a function of his more prominent role in the party. (Editing by Alistair Bell and Cynthia Osterman)

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